There will certainly be a variety of reasons that OS X will prompt for user credentials when attempting to send files to the trash, but there is one reason in particular that I see more frequently than others: It is the result of moving files from one OS X home directory to another.
Mac OS X uses Access Control Lists (ACLs) to manage file access. Each home directory has its own top-level set of ACLs. When a file is moved from one user’s home directory to another, the default ACLs in the new home directory may differ from the old. In my experience this may result in the new user account being able to successfully work with the file, but they are prompted for a password whenever they attempt to move it to the trash.
To fix this, you simply need to remove the ACLs from the files/folders you need prior to moving them. Once they are moved into the new home directory, the new and proper top-level ACLs will be inherited.
To remove ACLs, first open Terminal.
Use this command to remove ACLs from individual files:
sudo chmod -N filename
Use this command to recursively remove ACLs from a folder:
sudo chmod -R -N